Won't happen, Argentina's claim to the islands is simply the fact that they're close, which means we need to reignite our claim to the French state, and that they held the islands about 180 years ago. Seeming to ignore the only reason Britain took control was because Argentina ignored the deal struck with the British and consequently ignored all formal protests.
Nothing will happen from this, the UN can't really do much without the security councils backing of which the UK is a permanent member so really it could veto most of the measures taken against it and Argentina is unlikely to go to war yet again as the current government is just whiny, not the Military Junta of 1982.